The probability to find a random sequence of digits.

Notice 1: I am not sure where to post this thread, but this seems like the best place to me.

Notice 2: It is amazing where my mind will wander when I'm not entirely engaged on the events before me. I've been thinking about this concept for the hour during church. It's been years since I've attended any math class. By no means am I gifted in mathematics. So what I state below will be entirely elementary. I'm certain there will be mistakes in my train of thought. Please excuse my lack of professionalism in stating my thoughts.

I remember in my high school stats class that there is a difference in saying, "It has a 0% chance of occurring" and "It is impossible to occur". It was explained to me that something having a 0% chance of occurring would be flipping a coin one million times, and receiving heads each time (The probability of 0% chance of occurring still exists on a bell curve). Where as, being in China and United States simultaneously is impossible (impossible does not exist on the bell curve). Keeping this thought in mind, I move onto my next point.

Assume any irrational number. For the sake of my argument, I am only interested in the decimal portion of any irrational number. I will choose Pi as a concrete example, as it is most familiar to me. What I am interested in, is the possibility of finding any random digits of any length of digits in Pi. I will scaffold my thoughts, leading into an interesting argument.

Step 1: What is the likelihood of finding any one random digit in Pi? Example: What is the likelihood of finding a 3 in Pi? Answer: 100%, because if I am giving enough digits, I will find it. It is the 9th term in Pi.

Step 2: What is the likelihood of finding any two random digits in Pi? Example: 33? Answer: 100%, because if I run through enough digits, I will still find 33 in Pi. It is the 23rd - 24th term in Pi.

Step 3: What is the likelihood of finding any three random digits in Pi? Example: 330? Answer: Still 100%, because there is any infinite amount of digits in Pi, so I will come across it at some point. It is the +300term in Pi, to many digits for me to count through.

Generalization Step: What is the likelihood of finding any 'n'-random digits in Pi? Answer: 100% because there is an infinite amount of digits in Pi.

When discussing the term of infinity, my mind starts to flip out. It is a hard concept for me to grasp. So the next steps of my argument may sound funny. Again, all this will be very elementary, so please bare with me!

Because I know that it is 100% possible to find any 'n'-random digits in Pi I can assume that I can find any infinite-set of random digits in Pi.

Because I know that I can find any infinite-set of random digits in Pi, and I know Pi is infinitely long, then I know infinity is a sub-set of infinity!

If I know that infinity is a sub-set of infinity, then that means Pi is a subset of Pi! What?!?!

If Pi is a sub-set of Pi, then that means Pi repeats.

To generalize my last statement: Every irrational number is a sub-set of any other irrational number (including itself), then, that means every irrational number repeats!

If an irrational number repeats, then it is not irrational.

So there is no such thing as an irrational number.

So how about it? Interesting? Really cool!?!?! or should I just try real hard to keep paying attention to the Father's sermon during mass?